Why does my depth render look so washed out despite the model having lots of depth?
The "Depth"-pass uses the camera clipping range as bounds for its values. 

So if your clipping range is much larger than the Z-dimension of your terrain, it will only occupy a small range of all possible values. 
As all possible values are mapped to a value between 0 and 255 (at least on your screen, depending on your output format bitdepth you may get a finer resolution), you won't see much in your heightmap unless the camera range and the dimensions of your scene match.